PRIMITIVE GROUPS HAVING TRANSITIVE SUBGROUPS OF SMALLER, PRIME POWER DEGREE t

The groups in the title are classified, provided they are not too highly transitive. Let G be a primitive permutation group on a finite set S of n points. In 1871, Jordan initiated the study of G under the additional assumption that there is a transitive subgroup H of degree m, where 1 < m < n; that is, H fixes n m points and is transitive on the remaining points. THEOREM 1. I f m is a prime power, and G is not n m + 1-transitive.. then G is one of the following groups in its usual 2-transitive representation: a collineation group of P G ( d 1 , q) containing PSL(d,q) , where d > 3; the fu l l collineation group o] AG(d,2), where d >= 3; or a Mathieu group M22, Aut(M22), M23, or M24. This result contains several recent theorems found in [2], [3] and [4]. PROOF. G is 2-transitive on S ([7, p. 32]). By 12, Sect. 6], we may assume that G is not 3-transitive. Let B be the complement of the given set of m points, so I B[ = k = n m. Let P be a Sylow subgroup of the pointwise stabilizer G(B) of B, so that P is transitive on S B. By [2, (3.6)], the distinct sets B g, g E G, form a design ~ whose lines have more than two points; moreover, if B is not a line o f ~ , then planes o f ~ are well-defined and G is transitive on the set of planes. Suppose first that B is a line o f ~ . Let O e G be such that B n B ~ is a point x. Since P is transitive on the lines # B on x, the stabilizer P , of B g in P has index t This research was supported in part by NSF Grant GP 37982X. Received January 1, 1974